Primarily, unlike other pages, this is essentially more for myself than the reader, in that here I list my library as according to subject matter, to more easily see the references of any one subject. Note that this purposefully excludes mathematics section per se, in that the main pages are effectively this listing, with the addition of related materials here.

The page, as of 2020, is still very much in a developmental stage, to be added to ad hoc.

Contents

1. Perception: Impossible Figures, Optical Illusion etc. I may split this as according to specific aspects

2. Geometric Dissections

3. Publications featuring William Kolomyjec (tilings)

4. Optical Illusions

5. Square Limit

1. Perception: Impossible Figures, Optical Illusion etc.

Boring, Edwin J. ‘A New Ambiguous Figure’. American Journal of Psychology, Vol. 42, 1930, pp. 444-445. (22 April 2020)

Ostensibly on ‘My Wife and My Mother-in-Law’ discussion, as drawn by the well-known cartoonist A. E. Hill, in Puck, of 1915. However, the work itself is only discussed, briefly, at the end of the paper! This paper is commonly stated in the literature. It also mentions a ‘puzzle-picture’. I am unfamiliar with Gudden’s ‘brain-and-babies’, of 1896. I looked for this on the web but did not find it. Quoted by Block and Yuker.

Botwinick, Jack. ‘Husband and Father-in-Law - A Reversible Figure’. American Journal of Psychology, Vol. 74, 1961, pp. 312-313. (22 April 2020)

‘Husband and Father-in-Law’, stated as the complement to (Hills?) ‘My Wife and My Mother-in-Law’. It is not stated if this is of Botwinick’s devising. Quoted by Block and Yuker.

Bradley, Drake R. and Heywood M. Petry. ‘Organizational determinants of subjective contour: The subjective Necker cube’. American Journal of Psychology, Vol. 90, June 1977, No. 2, pp. 253-262 (23 April 2020)

Largely popular. Quoted by Block and Yuker.

Erb, Margaret B. and Karl M. Dallenbach. ‘‘Subjective Colors’ from Line-Patterns’. American Journal of Psychology, Vol. 52, June 1939, pp. 227-241 (23 April 2020)

Largely popular. Quoted by Block and Yuker.

Fisher, G. H. ‘Mother, Father and Daughter: A Three-Aspect Ambiguous Figure’. American Journal of Psychology, Vol. 81, June 1968b, pp. 274-277 ( April 2020)

Gillam, Barbara. ‘Geometrical Illusions’. Scientific American, Vol. 242, No. 1 January 1980, 102-111 (29 April 2020)

Generic.

Maheux, Magloire et al. ‘Geometrical Factors in Illusions of Direction’. The American Journal of Psychology, Vol. 73, No. 4, December 1960, pp. 535-543 (29 April 2020)

On Wundt and Zollner.

Kolers, Paul A. ‘The Boys From Syracuse: Another Ambiguous Figure’. American Journal of Psychology, Vol. 77, 1961, pp. 671-672. (22 April 2020)

Figure devised by ‘eminent caricaturist Al Hirschfeld’, in the Drama section of The New York Times, of 29 September 1963. Quoted by Block and Yuker.

Nicolas, Serge. ‘Joseph Delbouef on visual illusions: A historical sketch’. American Journal of Psychology Winter 1995, Vol.108, No. 4, 563-574 (29 April 2020)

Orbison, W. D. ‘Shape as a Function of the Vector Field’. American Journal of Psychology, 1939, Vol. 52, 1961, pp. 31-45. (27 April 2020)

Quoted by Block and Yuker.

Pierce, Andrew H. ‘Geometrical Optical Illusions’. Science, New Series Vol. 8, No. 206 December 9 1898, 814-829. (29 April 2020)

Ramachandran, Vilaynur S. ‘Perceiving Shape from Shading’. Scientific American 1988, August 259, 2, 76-83 (27 April 2020)

Quoted by Block and Yuker.

Sheon, Aaron. ‘Multistable Perception in Romantic Caricatures Studies in Romanticism’, Vol. 16, No. 3, Romanticism and Science (Summer, 1977), pp. 331-335. Published by: The Johns Hopkins University Press (6 May 2020)

From a reference in Dilworth.

Walker, Jearl. ‘What Explains subjective-contour illusions, those bright spots that are not really there?’ Scientific American, Vol. 258, No. 1 (January 1988), pp. 96-99 (27 April 2020)

Quoted by Block and Yuker.

Wilson, Amanda. ‘Multistable Perception of Art-Science Imagery’. Leonardo, Vol. 45, No. 2, 2012, pp. 156-164, 121 (6 May 2020)

Only the first two pages are of real interest, with subsequent pages ‘peripheral’.

2. Geometric Dissections

Cadwell, J. H. ‘Some Dissection problems Involving Sums of Cubes’. The Mathematical Gazette, Volume 48, No. 383, February 1964, pp. 391-396 (11 June 2020)

From a reference in Dissections: Plane & Fancy. Of no practical use.

Cadwell, J. H. 3278. ‘A Three-way Dissection Based on Ramanujan’s Number’. The Mathematical Gazette, Volume 54, February 19704, pp. 385-387 (11 June 2020)

From a reference in Dissections: Plane & Fancy. Of no practical use.

Cheney, William Fitch, Jr. Solution E4 (1932, 489) ‘Problems and Solution’. American Mathematical Monthly, Volume 40, p. 509-510) 1935 (11 June 2020)

From a reference in Dissections: Plane & Fancy. Of no practical use.

Cundy, H. M and C. D. Langford. 2875. ‘On the dissection of a regular polygon into n equal and similar polygons’. The Mathematical Gazette, Volume 44, No. 347, February 1960, p. 46 (11 June 2020)

From a reference in Dissections: Plane & Fancy. Cundy and Langford mentions Freese’s work here.

Goldberg, Michael. ‘Six-Piece Dissection of a Pentagon into a Triangle’. Problem E 972. American Mathematical Monthly, Volume 56, February 1940, p. 106-107 (11 June 2020)

From a reference in Dissections: Plane & Fancy.

Moser, Leo. ‘A Volume Dissection’. American Mathematical Monthly, Volume 56, December, p. 694 1949 (11 June 2020)

From a reference in Dissections: Plane & Fancy. Of no practical use.

Rosenbaum, Joseph. ‘A Dodecagon Dissection Puzzle’. Problem E 721. American Mathematical Monthly, Volume 54, p. 44 (11 June 2020)

From a reference in Dissections: Plane & Fancy. Of no practical use.

Wheeler, A. H. Problem E4. American Mathematical Monthly, Volume 42, ?, p. 509-510 (11 June 2020)

From a reference in Dissections: Plane & Fancy. Of no practical use.

3. Publications featuring William Kolomyjec (tilings)

Kolomyjec is of historical interest in regards of his two Escher-inspired works of Square Limit or Circle Limit. To this end, I examine various publications in which his work has appeared to see if they appear. Where not, this acts as a negative ‘seen and noted’.

Judy, Stephen N. Editor? ‘Some brief notes on computer art with notes on teaching computer art’

The English Journal. Published by the National Council of Teachers of English. Vol. 65, No. 1, Resources for 1976, pp. 1-105 (35 pages) (6 November 2020)

A ‘compilation’ of sorts? Cover and art, pages 15, 82: Computer-generated imagery by Bill Kolomyjec Engineering Instructional Services, Michigan State University

No Square Limit or Circle Limit type tilings.

Helmick, Richard. ‘Enhancing Creativity in Art and Design through Stochastically Generated Computer Graphics’. Published by the National Art Education Association. Art Education, Jul., 1984, Vol. 37, No. 4 (Jul., 1984), pp. 36-39 (6 November 2020)

No Square Limit or Circle Limit type tilings. Brief discussion/caption on Kolomyjec with ‘Bird Curves’, p 38-39:

Computer artist Kolomyjec, for example, may use a random number generator to choose locations of circles within a design and deterministically fix the points of squares. In another work, he may use random numbers to fill parameters in curve-generating equations. Mr. Kolomyjec claims some of his programs could be run a million times producing dramatically unique graphic output each time (Kolomyjec, 1977)

Kolomyjec, William J. (1977). Random technique and computer fine art. Waterloo, Ontario: Third International Conference on Computing in the Humanities

NOT SEEN

Leavitt, Ruth (ed). Artist and Computer. Harmony Books, 1976 (First saw on Internet Archive 19 October 2020)

From a reference in John L. Graham’s thesis (on Escher) and which incidentally started my interest in Kolomyjec (I may have seen his work previously, but not the Escher-like element). A collection of essays by the (35) leading lights on computer art of the day. William J. Kolamjec has a brief mention of Escher on pp. 48, 50-51, and shows a curious Square Limit and Circle Limit-type line drawings on p. 50, Fig. 8 ‘Homage to Escher’ and Fig. 9 ‘Escher in the Round’, of more than one tile, but quite what the artist is doing here is unclear. I lack the will and time to more fully investigate. Certainly, it is not a conventional self-similarity type.

Skimmed viewed all of the book (of a one-hour loan!) but there is nothing else of direct interest, such as proto parquet deformations.

Zientara, Marguerite. ‘Concept between ‘shotgun art’ approaches 1940s abstract expressionism’. Computerworld Vol. 12, No. 23, 5 Jun 1978, pp. 30-31. GOOGLE BOOKS (6 November 2020)

Nice discussion on Kolomyjec work in the round. No Square Limit or Circle Limit type tilings.

https://books.google.co.uk/books?id=ICoe1vr9x3kC&pg=PA30&lpg=PA30&dq=Kolomyjec,+William+J.++Random+technique+and+computer+fine+art.&source=bl&ots=jtwbhSMiHi&sig=ACfU3U3W-Cnd0HirOOUTbBjf-Em9kW4S9Q&hl=en&sa=X&ved=2ahUKEwiy2duZju7sAhWYilwKHW1KCNsQ6AEwEHoECAcQAg#v=onepage&q=Kolomyjec%2C%20William%20J.%20%20Random%20technique%20and%20computer%20fine%20art.&f=false

Computers and People. July-August, 1987 Vo l. 36, Nos. 7-8

Cover, Square Limit-type Angels and Devils

Computers and People. Sept.-Oct., 1980 Vol. 29, Nos. 9-10 p. 4, 15

formerly Computers and Automation

A Nike Reaching for a Grape

Bill Kolomyjec

Kolomputer Design

560 Pacific Pkwy.

Lansing, Ml 48910

This was executed on an Apple 11 Personal computer driving a Houston Instrument X-Y Digital Plotter. I am experimenting with the capabilities of these 3mall microprocessor graphic systems.

The Nike was digitized by hand from a drawing originally done on graph paper.

Linear interpolation· was used to generate the hair from the top and bottom curves which themselves were generated by algorithms. The bunch of grapes and "aura" were produced by a subroutine for drawing arcs, with an option for dashed lines.

The Lansing Star, May 20, 1977, p. 10. Art Fairs Guide. (6 November 2020)

Minor mention as a ‘participating artist’ in local East Lansing Art Festival exhibition. No Square Limit or Circle Limit type tilings.

Also see:

https://www.atariarchives.org/artist/sec15.php

Shows ‘Homage to Maurits C. Escher’ (Square Limit type) and ‘Escher in the Round’ (Circle Limit type)

4. Publications featuring Optical Illusions

Stewart, Ian. ‘Now you see it, now you don’t. What optical illusions tell us about our brains’. New Statesman, pp. 36-41, 20 December 2013- 9 January 2014

Escher mention in passing p. 38, Relativity, p. 41.

5. Publications featuring Square Limit

Books and Articles

Anon. M. C. Escher Universe of Mind Play, pp. 132, 153-163, 166, 170

Print p. 132; numerous sketches but no discussion, pp. 153-163, 166. Cancelled wood blocks, p. 170

Bell, Marc. The Magical World of M.C. Escher, p. 164.

Just the print and details, no apparent discussion.

Chailloux, Emmanuel and Guy Cousineau. Programming Images in ML. Proceedings of the ACM SIGPLAN Workshop on ML and its Applications (1992). NOT SEEN

Coxeter, H. S. M. et al. M.C. Escher: Art and Science, pp. 57-60.

Grunbaum, Branko. ‘Mathematical Challenges in Escher's Geometry’, pp. 53-67

Square Limit discussion pp. 57-60

Ernst, Bruno. The Magic Mirror of M. C. Escher, pp. 103-105.

An in-depth discussion by Ernst, the best of what little there is.

Escher, M. C. Regelmatige vlakverdeling

See Wilson, which reprints this.

Escher, M. C. The Graphic Work of M. C. Escher, p. 15.

Brief commentary of a few lines of just basic detail, print, No. 25

Henderson, P. Functional geometry. In Conference Record of the 1982 ACMSymposium on Lisp and Functional Programming, D.P. Friedman and D.S. Wise (Eds.). Pittsburgh, Pennsylvania, ACM Press, 1982, pp. 179–187.

See http://www.ecs.soton.ac.uk/~ph/funcgeo.pdf.

Locher, J. L. The World of M. C. Escher, pp. 256-257.

Preparatory drawings, p. 256 and print, p. 257, but no discussion.

Locher, J. L. (editor) The Infinite World of M. C. Escher, p. 148.

Print, p. 148, no discussion.

Locher, J. L. et al. Escher The Complete Graphic Work, pp. 118, 325, 343.

Bool catalogue 443. p. 325, p. 343 (no other apparent references) Square Limit type division diagram in p. 169, in reprint of Regelmatige vlakverdeling, published 1958

Schattschneider, Doris. Visions of Symmetry, pp. 252-253.

Letter to H. S. M. Coxeter, 1964, p. 253. A relatively in-depth discussion.

Schattschneider, Doris. ‘Coxeter and the Artists - Two-way Inspiration’. In The Coxeter Legacy: Reflections and Projections pp. 262-280. 2006. Edited by Harold Scott Macdonald Coxeter, Chandler Davis, Erich W. Eller. Fields Inst. Comm. ser. no. 46, Amer. Math. Soc., 2005

Primarily of a detailed discussion on Coxeter and Circle Limits, with a brief, in passing references to Square Limit p. 266 with an illustration and brief discussion, possibly continued, but curtailed in Google Books! However, it seems unlikely to be extensive.

Schattschneider, Doris. ‘Escher’s Metaphors’. Scientific American, Vol. 271, No. 5 November 1994, pp. 66-71

Teuber, Marianne L. 'Sources of Ambiguity in the Prints of Maurits C. Escher.' Scientific American 231 No. 1 (July 1974): 90-104.

See pp. 89.

Thé, Erik (designer). The Magic of M. C. Escher, p. 182.

Large scale print, p. 182, and accompanying letter to Gerd Arnzt, 14 April 1964 (quoted by Ouyang), and detail, p. 183, but no discussion as such.

Wilson, Janet, (editor). Escher on Escher, p. 41.

Brief reference to Square Limit, p. 41, in prepared notes of Escher’s proposed 1964 US lecture. A minor discussion in passing, in context of infinity, with Development II, Path of Life I, Whirlpools, Sphere Surface with Fish, Smaller and Smaller, Square Limit, Circle Limit IV, and Circle Limit III. In short, inconsequential.

Not in (or believed to be):

Jeffrey Price Amazing Images

Letters to Canada

M.C. Escher’s Legacy

An open question is to when Escher devised the ‘square limit’ process. This can be dated as of least, and likely, of October 1956, where Smaller and Smaller I shows the process later used in a different form in Square Limit.